This is a generalisation of the commuting graph of a group, which has the elements of a group as vertices, joined by an edge if they commute see . In this paper we will consider a generalisation of the graph of Herzog, Longobardi and Maj. For a finite group G we define the graph T G to be the graph whose vertices are the conjugacy.
Departament d' Algebra, Universitat de Valencia, Dr. Moliner, 50, Burjassot, Valencia, Spain e-mail: Adolfo.
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Ballester uv. Cossey anu. We characterise those groups G for which F G is complete. Permutability, like normality, is not a transitive relation in general.see
Groups satisfying certain rank conditions | Dixon | Algebra and Discrete Mathematics
According to a classical result of Ore  permutable subgroups of finite groups are subnormal. Hence a finite group is a PT-group if and only if every subnormal subgroup is permutable. Suppose that G is a soluble PT-group. In the other direction, we argue by induction on the order of G. We begin by showing that G is a soluble PT-group if G has at least two minimal normal subgroups. Thus we suppose that all minimal normal subgroups have the same prime order p. Thus all p-chief factors are G-isomorphic.
Therefore G is supersoluble and all chief factors of the same order are G-isomorphic. If the p-chief factors are central, then G is a p-group with all proper quotients modular and so is itself modular, since by Theorem of Longobardi  such a group must have a unique minimal normal subgroup. Assume now that MN is a Sylow p-subgroup of G. Consequently M and N are G-isomorphic. We now suppose that G has a unique minimal normal subgroup N. Let p and q be different primes dividing the order of Si and let xi and yi be elements of Si of orders p and q, respectively.
Then xi.. The projection of xi A result of Abe and Iiyori [i] shows that this is impossible. Consequently N is a p-group for some prime p. Let Q be a cyclic normal subgroup of M. It now follows that every element of F acts as a power automorphism on N and hence F acts as a power automorphism group on N. Since power automorphisms are central in the power automorphism group of N [5, Theorem 2. Thus M is a nilpotent modular group. In particular M is a p'-group.
Let Q be a non-abelian Sylow q-subgroup of M. By Iwasawa's Theorem [ii, Theorem 2.
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In both cases every cyclic subgroup of Qo is normal in Q and hence in M. Let U be a cyclic subgroup of N and let R cyclic subgroup of Qo. By hypothesis, there exits an element a e M such that RUa is subgroup.
This implies that Q normalises U and Q acts as power automorphisms on N. It now follows that M acts as power automorphisms on N and so N is a cyclic group of order p and M is cyclic of order dividing p — i and G is clearly a PT-group. Now suppose that N is contained in the Frattini subgroup of G. Assume that G is not modular. By the Theorem of Longobardi  either G is the central product of a subgroup P isomorphic to M p and another subgroup or G is isomorphic to.
In the first case it is clear that if P is generated by a and b of order p no conjugate of a will commute with b and hence will not permute with b. Since w centralises b and b has order at least 4, w acts as universal power automorphism on H by a theorem of Napolitani [11, Theorem 2. Thus G cannot be nilpotent. Then there exists a p'-subgroup D of G complementing E in G.
It follows that D is cyclic of order dividing p — 1.
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Assume that Ep is not trivial. Chapter 2 Modular lattices and abelian groups.
Chapter 6 Projectivities and normal structure of infinite groups. Chapter 7 Classes of groups and their projectivities.
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Chapter 3 Complements and special elements in the subgroup lattice of a group. Chapter 4 Projectivities and arithmetic structure of finite groups. Chapter 5 Projectivities and normal structure of finite groups. Chapter 8 Dualities of subgroup lattices. Chapter 9 Further lattices. Index of Names. Index of Subjects.